"Reproducibility" of the Asymptotics of ODEs

Question

I asked this question here but got no reply.

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My question has its origin and is related to this problem in heat conduction but is more general in scope.

Let $\mathbf {f(x, u)}$ be a sufficiently smooth function of variables $\mathbf x \in \mathcal D_x \subset \mathbb R^n$ and $\mathbf u \in \mathcal D_u \subset \mathbb R^m$ such that

1. $\mathbf {f(0, 0) =0}$;
2. for a given fixed vector $\mathbf u=\mathbf u \in \mathcal D_u$ the equation $\mathbf {f(x, a)}=0$ is solved by $\mathbf {x =x_a}\in \mathcal D_x.$

Define a positive causal monotonic smooth function $p(t)$ as follows $p(t) = 0 \text{ for } t<0$ and $p(t) =1 \text{ for } t>\tau_0 \ge 0$ for some fixed risetime $\tau_0$. If using the Heaviside step function then $\tau_0 = 0.$

Now consider the following autonomous differential equation: $$ \dot {\mathbf {x}} = \mathbf {f(x, u)} \tag{1}$$ where the excitation vector is defined as $$ \mathbf {u} = [a_1p(t-\theta_1), a_2p(t-\theta_2),..., a_mp(t-\theta_m)] \tag{2}.$$ And, finally let us also assume that Eq. 1 has a solution for arbitrary amplitude vector $\mathbf a \in \mathcal D_u$ and starting values $0\le \theta_k \le \theta_0; k=1,2,..,m$ for some fixed $0 \le \theta_0$ so that asymptotically as $t\to \infty$ we have $\mathbf x(t) \to \mathbf {x_{a,\theta}}$ in which I have indicated that the asymptotic values of the solution may also depend on the starting instances $\theta_k$.

I am looking for a reasonably "large" set of function $\mathbf f$ for which we can state that under these conditions the asymptotic limit is independent of the values $\theta_k \le \theta_0$ if the fixed $\theta_0$ is sufficiently small. Ideally, I am hoping for a criterion stating that the order of switching in the excitations does not matter if the system is, say, Lyapunov stable or something like that.

If the matrix $\mathbf A$ is Hurwitz stable having all eigenvalues with negative real part, then for the linear equation of constant coefficients $\mathbf {\dot x = Ax + Bu}$ this type of "stability", or asymptotic independence on the starting instances, is true because all the transients will decay exponentially, but I would like to know if it can be shown to hold in some non-linear case as well. A simple perturbation scheme on the excitation does not seem to work because the amplitudes can be arbitrarily large.

Answer 1

It seems to me that what happens in the linear case is that $\dot x=Ax+Ba$ has a (unique) globally asymptotically stable equilibrium, namely $x=-A^{-1}Ba$. This will be the limit independently of $\theta$. The conclusion will be the same as soon as $\dot x=f(x,a)$ has a globally asymptotically stable equilibrium.